Lesson 7 Combinational Logic Circuits Concepts Introduced Adders - PowerPoint PPT Presentation

Lesson 7 Combinational Logic Circuits

Concepts Introduced • Adders • Multiplexers • Decoders • basic Arithmetic/Logic Unit

Review Half-Adder • A half adder takes two inputs, a and b, and generates two outputs, the carry and the sum. • A half adder is called a (2,2) adder as it takes two inputs and produces two outputs. • The circuit for the carry can use an AND gate. • The circuit for the sum can use an XOR gate.

Truth Table for a Half-Adder

Half-adder

Full-adder • Full-adder is composed of two half-adders and an OR gate. • The full-adder is a three input and two output combinational circuit. • The first two inputs are A and B and the third input is an input carry as C-IN. • The output carry is designated as C-OUT and the normal output is designated as S which is SUM. • Therefore, a full adder adds binary numbers and accounts for values carried in as well as out • A full adder adds binary numbers and accounts for values carried in as well as out.

Full-adder

A Logic Diagram for a Full-Adder

A Truth Table for a Full-Adder Sum = x’.y’.CarryIn + x’.b.CarrryIn’ + a.b’.CarryIn’ + a.b.CarryIn CarryOut = x’.y.CarryIn + x.b’.CarrryIn + a.b.CarryIn’ + a.b.CarryIn

Ripple-carry adder • We can be able to create a logical circuit using multiple adders to add N bit numbers. • Each full adder inputs a C in , which is the C out of the previous adder, • we can build an adder capable of adding two 16-bit words, for example, by replicating the above circuit 16 times, feeding the Carry Out of one circuit into the Carry In of the circuit immediately to its left. • This circuit is called a ripple-carry adder

Ripple-Carry Adder

Adders Summary • Adders are very important circuits—a computer would not be very useful if it could not add numbers • Another important operation that all computers use often is decoding binary information from a set of n inputs to a maximum of 2 n outputs.

Decoder • A decoder uses the inputs and their respective values to select one specific output line. By selecting we mean that one unique output line is asserted, or set to 1 while the other output lines are set to zero. • Decoders are defined by the number of inputs and the number of outputs. a decoder that has 3 inputs and 8 outputs is called a 3-to-8 decoder. • All memory addresses in a computer are specified as binary numbers. When a memory address is referenced, the computer first has to determine the actual address. This is done by using a decoder

Decoders Continued • The most common type of decoder has an n -bit input and 2 n outputs, where only one output is asserted for each input combination. • This decoder translates the n -bit input into a signal that corresponds to the binary value of the n -bit input.

A 3-bit decoder has 3 inputs, called 12, 11, and 10, and 2^3 = 8 outputs, called Out0 to Out7

Decode and Decoder Symbol (left to right)

Mul Multi tipl plexors • A multiplexor is also called a selector , since its output is one of the inputs that is selected by a control. • Multiplexors can be created with an arbitrary number of data inputs. • When there are only two inputs, the selector is a single signal that selects one of the inputs if it is true (1) and the other if it is false (0) • If there are n data inputs, there will need to be ⌈ log 2 n ⌉ selector input

Parts of a multiplexor • A decoder that generates n signals, each indicating a different input value • An array of n AND gates, each combining one of the inputs with a signal from the decoder • A single large OR gate that incorporates the outputs of the AND gates

A Look Inside a Multiplexer and A Multiplexer Symbol

two-input multiplexor on the left and its implementation with gates on the right

Arithmetic Logic Unit (ALU) • We have discussed enough combinational circuit to build an ALU • The arithmetic logic unit (ALU) carries out the logic operations (such as comparisons) and arithmetic operations (such as add or multiply) required during the program execution. • the device that performs the arithmetic operations like addition and subtraction or logical operations like AND and OR. • Generally an ALU has two data inputs and one data output. • Operations performed in the ALU often affect bits in the status register • The ALU knows which operations to perform because it is controlled by signals from the control unit.

The symbol to represent an ALU

1-bit logical unit for AND and OR

1-bit logical unit for AND and OR • In the 1-bit logical unit for AND and OR, the multiplexor on the right then selects a AND b or a OR b , depending on whether the value of Operation is 0 or 1 • Both the AND and OR operations are always performed, but the output produced depends on the selector to the multiplexor • A 32-bit logical unit for AND and OR operations would just be an array of these 1-bit logical units.

ALU • We next include addition to the ALU • An adder must have two inputs for the operands and a single-bit output for the sum. There must be a second output to pass on the carry, called CarryOut . • We also need a third input. This input is called CarryIn .

Full-adder

Input and output specification for a 1-bit adder

CarryOut from the Full Adder

A 32-bit ALU constructed from 32 1-bit ALUs

A 1-bit ALU that performs AND, OR, and addition

A 1-bit ALU that performs AND, OR, and addition on a and b or not a and not b

Re Reading • Hennessy and Patterson Chapter 8.3 and 8.5 (Appendix B)